Is there any problem with keeping the monitor on all the time (even at night when no one is in the office)?
It seems to me there must be (besides electricity usage).
Does it burn the image onto he screen.
I have a regular monitor.
Another person in the office has a new flat screen monitor.

I've been told to turn them off and to turn them off, which should I do?

The idea that your monitor is 'on' is from the old days, new monitors are basically completely asleep, that's what the little orange indicator means, it's not really powered up, in terms of energy consumption when the monitor is in its sleep state it probably is using less energy than any other single thing in your office.

That's what those 'power saver' features are about, they have the main power turn off in the monitor when you don't use the computer, that means that the monitor isn't really on per se, until you hit a key, then it gets a little signal, and that signal actually turns it back on, full energy mode.

There is never a need to physically turn off the on/off button on a monitor or a printer, that 's all automatic now.

Worry about leaving lights on, that eats up probably 100 times more energy than the monitor does in its sleep state.

However, you shouldn't leave the monitor totally on, that's bad, yes it can burn the image into the monitor, and yes it is a total waste of electricity, and yes it will dramatically shorten the life of the monitor, when it's on, it's being used, and electrical stuff wears out the more it gets used.

So make sure you have the power saving feature turned on, that's the thing that shuts off the monitor when it's inactive for x number of minutes.

You can find that here:
Control Panel -> Power Options -> Power Schemes.

For desktops, choose 'Home office/desktop

Select
Turn off monitor - 20 minutes, or however long you want.
turn off harddrives - never [I've had drives fail using that feature]
go into standby - never [too many bugs]

Then you just tap a key, and the monitor starts.

The same goes for leaving the desktop running at night, this is a waste of money and electricity, and Windows needs to be rebooted to stay working decently for most users, turn off your computer when you leave the office, turn it on when you come back in the morning.